Trigonometric display and method thereof

ABSTRACT

A trigonometric function display clock includes a clock face that has a center aligned with a central axis, axial markings aligned with an x-axis and a y-axis, and a secondary circle having a diameter equal to a radius of the axial markings. A first point on the secondary circle is aligned at the center of the central axis, and the secondary circle is configured to be rotatable about the first point. The secondary circle is configured to overlap and align with at least a portion of the axial markings as the secondary circle rotates about the first point.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 62/578,989 filed on Oct. 30, 3017 and entitled “TrigonometricDisplay and Method Thereof,” which is incorporated herein in itsentirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

REFERENCE TO A MICROFICHE APPENDIX

Not applicable.

BACKGROUND

Mathematical functions are universally used in a variety of industries.In order to understand various mathematical functions, a variety ofvisualizations can be performed. For example, triangles and other shapesare used to depict trigonometric functions. While these visualizationsare useful, additional ways of depicting such functions may helpcommunicate the information.

SUMMARY

In an embodiment, a trigonometric function display clock comprises aunit circle disposed on a clock face that has a center of the unitcircle aligned with a central axis, axial markings aligned with anx-axis and a y-axis, and a secondary circle having a diameter equal to aradius of the unit circle. A first point on the secondary circle isaligned at the center of the unit circle, and the secondary circle isconfigured to be rotatable about the first point. The secondary circleis configured to overlap and align with at least a portion of the axialmarkings as the secondary circle rotates about the first point. Theclock can have at least one hand configured to rotate about the centralaxis, and the at least one hand can be an hour hand, a minute hand, anda second hand. The secondary circle can be coupled to the at least onehand. The secondary circle can be disposed within the unit circle, and asecond point on the secondary circle can align with the unit circle.

In an embodiment, a trigonometric function display clock comprises aunit circle disposed on a clock face, where a center of the unit circleis aligned with a central axis, axial openings in the clock face thatare aligned with an x-axis and a y-axis, and a secondary circular dischaving a diameter equal to a radius of the unit circle. A first point onthe secondary circular disc is aligned at the center of the unit circle,and the secondary circular disc can be configured to be rotatable aboutthe first point. The secondary circular disc can be disposed on a firstside of the clock face. The secondary circular disc can be visiblethrough the axial openings from a second side of the clock face. Theclock can also include at least one hand configured to rotate about thecentral axis, and the at least one hand can be disposed on the secondside of the clock face. The secondary circular disc can be coupled tothe at least one hand, and the at least one hand can comprise an hourhand, a minute hand, and a second hand. A second point on a perimeter ofthe secondary circular disc can align with the unit circle.

In an embodiment, a method of displaying one or more trigonometricfunctions comprises: providing a display device comprising a unit circledisposed on a clock face, wherein a center of the unit circle is alignedwith a central axis, axial markings aligned with an x-axis and a y-axis,and a secondary circle having a diameter equal to a radius of the unitcircle. A first point on the secondary circle is aligned at the centerof the unit circle, and a second point on a perimeter of the secondarycircle aligns with the unit circle. The method can also include rotatingthe secondary circle about the first point, where the secondary circleintersects with at least a portion of the axial markings as thesecondary circle rotates about the first point, and determining at leastone of a value of a sine function or a value of a cosine function at thesecond point based on the overlap between the secondary circle and theaxial markings. Determining the at least one of the value of the sinefunction or the cosine function at the second point can comprisedetermining a value of the cosine function at the second point, wherethe value of the cosine function corresponds to a proportional lengthof: 1) a first length of the axial marking between the central axis andan intersection between the secondary circle and the axial marking onthe x-axis, and 2) a total length between the central axis and the unitcircle along the axial marking on the x-axis. Determining the at leastone of the value of the sine function or the cosine function at thesecond point can comprise determining a value of the sine function atthe second point, where the value of the sine function corresponds to aproportional length of: 1) a first length of the axial marking betweenthe central axis and an intersection between the secondary circle andthe axial marking on the y-axis, and 2) a total length between thecentral axis and the unit circle along the axial marking on the y-axis.

In some embodiments, the axial markings comprise openings in the clockface, and the secondary circle comprises a perimeter of a secondarycircular disc. The secondary circular disc can be disposed on a firstside of the clock face, and the secondary circular disc is visiblethrough the openings from a second side of the clock face. In theseembodiments, determining the at least one of the value of the sinefunction or the cosine function at the second point can comprisedetermining a value of the cosine function at the second point, wherethe value of the cosine function corresponds to a proportional lengthof: 1) a first length of the secondary circular disc between the centralaxis and the unit circle visible through the opening on the x-axis, and2) a total length between the central axis and the unit circle along theopening on the x-axis. Similarly, determining the at least one of thevalue of the sine function or the cosine function at the second pointcan comprise determining a value of the sine function at the secondpoint, where the value of the sine function corresponds to aproportional length of: 1) a first length of the secondary circular discbetween the central axis and the unit circle visible through the openingon the y-axis, and 2) a total length between the central axis and theunit circle along the opening on the y-axis. The display device can alsoinclude at least one hand configured to rotate about the central axis,and the at least one hand can be an hour hand, a minute hand, and asecond hand. The secondary circle can be coupled to the at least onehand. The method can also include determining a time using the at leastone hand on the display device. The display device can also include anenclosing square displayed around the unit circle, and the method canalso include determining at least one of a tangent value or a cotangentvalue at the second point using the enclosing square.

These and other features will be more clearly understood from thefollowing detailed description taken in conjunction with theaccompanying drawings and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present disclosure, referenceis now made to the following brief description, taken in connection withthe accompanying drawings and detailed description, wherein likereference numerals represent like parts.

FIG. 1 schematically illustrates a clock face according to anembodiment.

FIG. 2 illustrates a front elevation view of a clock face according toan embodiment.

FIG. 3 illustrates a front elevation view of another clock faceaccording to another embodiment.

FIG. 4A illustrates an exploded view of an embodiment of the clock faceshown in FIG. 3.

FIG. 4B illustrates an exploded view of another embodiment of the clockface shown in FIG. 3.

FIG. 5 schematically illustrates another clock face according to anotherembodiment

FIG. 6 schematically illustrates a processor and associated system thatcan be used to provide the clock face in some embodiments.

DETAILED DESCRIPTION

It should be understood at the outset that although illustrativeimplementations of one or more embodiments are illustrated below, thedisclosed systems and methods may be implemented using any number oftechniques, whether currently known or not yet in existence. Thedisclosure should in no way be limited to the illustrativeimplementations, drawings, and techniques illustrated below, but may bemodified within the scope of the appended claims along with their fullscope of equivalents.

The following brief definition of terms shall apply throughout theapplication:

The term “comprising” means including but not limited to, and should beinterpreted in the manner it is typically used in the patent context;

The phrases “in an embodiment,” “according to one embodiment,” and thelike generally mean that the particular feature, structure, orcharacteristic following the phrase may be included in at least oneembodiment of the present invention, and may be included in more thanone embodiment of the present invention (importantly, such phrases donot necessarily refer to the same embodiment);

If the specification describes something as “exemplary” or an “example,”it should be understood that refers to a non-exclusive example;

The terms “about” or “approximately” or the like, when used with anumber, may mean that specific number, or alternatively, a range inproximity to the specific number, as understood by persons of skill inthe art field; and

If the specification states a component or feature “may,” “can,”“could,” “should,” “would,” “preferably,” “possibly,” “typically,”“optionally,” “for example,” “often,” or “might” (or other suchlanguage) be included or have a characteristic, that particularcomponent or feature is not required to be included or to have thecharacteristic. Such component or feature may be optionally included insome embodiments, or it may be excluded.

Disclosed herein is a display that allows various trigonometricfunctions to be visualized. The display can be used as a teaching toolor decorative display. In some instances, the display can be combinedwith a clock face so that the resulting device can serve as both a clockand a visual indicator of the trigonometric functions.

As an example of a display, a clock can comprise the various hands suchas the hour hand, the minute hand, and optionally, the second hand. Afirst circle can be created around and centered on the rotational axisof the time hands, where the first circle can have a first diameter. Thefirst circle can generally be stationary and would be placed at or neara perimeter of the clock face. A second circle can then be placed sothat an edge of the second circle is aligned along and rotates about theaxis of rotation of the time hands. The second circle can have adiameter that is exactly half that of the first circle. The secondcircle can rotate with the time hands about the axis of rotation of thetime hands, and an outer diameter of the second circle can then alignwith the first circle. Two perpendicular cross lines can be created onthe face of the display. As the second circle rotates about the axis ofthe time hands, the second circle will intersect with the cross lines,thereby creating a line that represents a value or magnitude of one ormore trigonometric functions, as described in more detail herein. Thus,the proposed displays can be used to visualize various functions whilein some instances also serving as a time piece.

In an embodiment, a clock face can be used to visually display varioustrigonometric functions. As shown in FIG. 1, the clock face 100 caninclude a unit circle 102 and a plurality of time hands including anhour hand 108, a minute hand 106, and a second hand 104. In someinstances less than all of the hands may be present. The unit circle 102can be used on or for the clock face 100. A secondary circle 112 can bealigned with at least one of the hands, where the secondary circle 112has a point aligned with the center of the hands. For example, the handscan be seen to rotate about an axis extending normal to the plane of theclock face 100 in FIG. 1. The secondary circle 112 can also rotate aboutthis axis such that an outer point on the secondary circle 112 movesalong the unit circle 102. Angular labels can optionally be included tomark the positions along the unit circle 102. For example, the angularpositions can be labeled in radians, and these positions can be labeledalong with or in place of the hour markings normally found on a clock.

As clock hands 104, 106, 108 rotate, they define angles that can be seenas distances along the edge of the unit circle 102 from a base position(e.g., a zero or 2π position). As shown in FIG. 1, the 3 o'clockposition can be a base position, and the angles can be positive ifcounter-clockwise or negative if clockwise, though the angles can bedefined in opposite directions as well. The unit circle 102 can have aradius 110. In an embodiment, the radius 110 of the unit circle 102 canbe defined, for example, by the length of the second hand 104 (generallytaken as the longest hand having a length that is greater than the hourhand 108 and the minute hand 106). In this example, attaching asecondary circle 112 to the second hand 104 whose diameter is the samelength as the second hand 104 yields the cosine value of the second hand104 angle as the secondary circle 112 intersects the x axis and the sinevalue of the second hand 104 angle as the secondary circle 112intersects the y-axis. The cosine value is the signed distance from theorigin to the x intersection point along the x-axis, yielding valuesbetween −1 and 1. Similarly, the sine value is the signed distance fromthe origin to the y intersection point along the y-axis, yielding valuesbetween −1 and 1. Thus, the use of the unit circle 102 with thesecondary circle 112 allows various mathematical functions to bedisplayed.

While described in terms of the second hand 104, any of the hands 104,106, 108 can be used with the secondary circle 112. In some embodiments,the lengths of the hands 104, 106, 108 may be modified from their usuallengths in order to allow for placement of the secondary circle 112 on ahand other than the second hand. For example, the hour hand 108 and/orthe minute hand 106 can be lengthened, and the seconds hand canoptionally be shortened, to allow for use of the secondary circle 112with the hour hand 108 and/or the minute hand 106.

The clock face 100 illustrates that any vector (e.g., as illustrated byone of the hands of the clock) can be represented as the sum of twoorthogonal arrows or vectors. The dynamic movement of the second handemphasizes how cosine and sine are parameterized by arc length.Labelling arc length with numbers rather than fractions of π may help toenforce the meaning of π being equal to approximately 3.1416 as thescale factor times a radius to get the length of a semicircle (or timesa diameter to get a full circle). The breakdown of the two vectors canbe taken as a visual estimate of the cosine and sine functions of arclength.

The clock display 100 can be created using a variety of technologies. Asshown in FIG. 1, the clock face 100 can be created by a processorexecuting an application stored in a memory, where the computerimplementation components are described in more detail herein. Theapplication can create the display of the clock 100, the hands 104, 106,108, the unit circle 102, and the secondary circle 112. When created asa computer display, functional indicators such as the cosine indicator122 and the sine indicator 124 can be provided to graphically illustratethe value of the resulting sine or cosine functions. In theseembodiments, the unit circle 102 and the secondary circle 112 areoptional as they indicators 122, 124 can be displayed graphicallywithout the need for the intersection of the unit circle 102 and thesecondary circle 112. Further, the hands can comprise different colorsto make the distinction between the clock hands in the display clearer.

The cosine indicator 122 can provide an indication of the cosine valuecorresponding to the intersection of the secondary circle 112 with thex-axis. The value of the cosine indicator can range between −1 to 1,where a zero line length corresponds to a value of zero. A line lengthbetween the center and the positive intersection at the unit circle 102(e.g., the 3 O'clock position) can be defined to have a value of 1, anda line length between the center and the negative intersection at theunit circle 102 (e.g., the 9 O'clock position) can be defined to have avalue of −1. Similarly, the sine indicator 124 can provide an indicationof the sine value corresponding to the intersection of the secondarycircle 112 with the y-axis. The value of the sine indicator can rangebetween −1 to 1, where a zero line length corresponds to a value ofzero. A line length between the center and the positive intersection atthe unit circle 102 (e.g., the 12 O'clock position) can be defined tohave a value of 1, and a line length between the center and the negativeintersection at the unit circle 102 (e.g., the 6 O'clock position) canbe defined to have a value of −1. The functional indicators candynamically change as the secondary circle 112 rotates around the clockface.

The clock display can also be created using a mechanical clock. As shownin FIG. 2, a mechanical clock face 200 can include the unit circle 102and the plurality of time hands including an hour hand 108, a minutehand 106, and a second hand 104. The unit circle 102 can be used on orfor the clock face 200. A secondary circle 112 can be aligned with atleast one of the hands, where the secondary circle 112 has a pointaligned with the center of the hands. Thus, the main components of theclock face 200 are generally similar to those presented in FIG. 1, andsimilarly numbered elements can be the same or similar to thosedescribed with respect to FIG. 1.

The mechanical clock face 200 can generally have a motor or othermechanism coupled to the hands 104, 106, 108 to create rotation aboutthe central axis over time. The clock face 200 can comprise a relativelyplanar sheet located between the motor and the hands 104, 106, 108,where the drive shafts extend through the clock face 200 and define therotational axis of the hands 104, 106, 108. In this configuration, themotor would reside on the opposite side of the clock face 200 from thehands 104, 106, 108.

The unit circle 102 can comprise a circular marking or device attachedto the clock face 200. For example, the unit circle 102 can be paintedor printed on the clock face 200 and/or a material such as disk or tubeof material (e.g., plastic, metal, glass, paper, etc.) can be attachedto the clock face 200 to form the unit circle 102. In the example shownin FIG. 2, a secondary circle 112 can be attached to the second hand104, where the secondary circle 112 has a diameter equal to a radius ofthe unit circle 102. An edge of the secondary circle 112 can be alignedwith the axis of rotation of the second hand 104 such that rotation ofthe second hand aligns an outer point of the secondary circle 112 withthe unit circle 102. The secondary circle 112 can be formed of a ring ofmaterial (e.g., metal, plastic, glass, paper, etc.) that is attached tothe second hand 104.

In addition to the unit circle 102 and the secondary circle 112, axialmarkings 202, 204 representing the y-axis and the x-axis, respectively,can be attached to the clock face 200. The axial markings 202, 204 canbe formed in the same manner as or similarly to the unit circle 102 onthe clock face 200. In use, the secondary circle 112 and the axialmarkings 202, 204 function to display the cosine and sine functions atthe intersection of the axial markings 202, 204 with the secondarycircle 112. For example, the cosine can be displayed by providing anindication of the cosine value corresponding to a length 210 of theaxial marking 204 between the rotational axis and the intersection ofthe secondary circle 112 with the axial marking 204 for the x-axis. Thevalue of the cosine can range between −1 to 1, where a zero length ofthe axial marking 204 within the secondary circle 112 corresponds to avalue of zero for the cosine. A length of the axial marking 204 withinthe secondary circle 112 between the center and the positiveintersection at the unit circle 102 (e.g., the 3 O'clock position) canbe defined to have a value of 1, and a length of the axial marking 204within the secondary circle 112 between the center and the negativeintersection at the unit circle 102 (e.g., the 9 O'clock position) canbe defined to have a value of −1. As shown in FIG. 2, the value of thecosine is illustrated by the relative amount of the axial marking 204within the secondary circle 112 between the rotational axis towards the3 O'clock position.

Similarly, the sine can be displayed by providing an indication of thesine value corresponding to a length 212 of the axial marking 202 forthe y-axis between the rotational axis and the intersection of thesecondary circle 112 with the axial marking 202 for the y-axis. Thevalue of the sine can range between −1 to 1, where a zero length of theaxial marking 202 within the secondary circle 112 corresponds to a valueof zero for the sine. A length of the axial marking 202 within thesecondary circle 112 between the center and the positive intersection atthe unit circle 102 (e.g., the 12 O'clock position) can be defined tohave a value of 1, and a length of the axial marking 202 within thesecondary circle 112 between the center and the negative intersection atthe unit circle 102 (e.g., the 6 O'clock position) can be defined tohave a value of −1. As shown in FIG. 2, the value of the sine isillustrated by the relative amount of the axial marking 202 within thesecondary circle 112 between the rotational axis towards the 6 O'clockposition.

As shown in the embodiment of FIG. 2, the display clock 200 can includeone or more hands 104, 106, 108 to illustrate time, a unit circle 102encircling the hands, axial markings for the x-axis 204 and the y-axis202, and a secondary circle 112 having a diameter equal to a radius ofthe unit circle 102. The secondary circle 112 can have a first point onthe circle aligned at the center point of the unit circle 102, and thesecondary circle 112 can be arranged to rotate about the first point. Inthis arrangement, an outer point on the secondary circle 112 aligns witha point on the unit circle 102. The sine and cosine functions can thenbe displayed based on an intersection of the unit circle with themarkings for the x-axis 204 and the y-axis 202. When used on a clockface 200, the secondary circle 112 can be coupled to one of the hands104, 106, 108 to rotate about the center point during use of the clock,thereby serving both to illustrate the time as well as the trigonometricfunctions of at least sine and cosine.

An alternative embodiment of a mechanical clock face is shown in FIGS. 3and 4A-4B, where FIG. 3 is a perspective view of the face of the clockwhile FIGS. 4A-4B are exploded views of embodiments of the same clock.As shown in FIGS. 3 and 4A-4B, a mechanical clock face 300 can includethe unit circle 102 and the plurality of time hands including an hourhand 108, a minute hand 106, and a second hand 104. The unit circle 102can be used on or for the clock face. In the embodiment of FIGS. 3 and4A-4B, the secondary circle is in the form of a secondary circular disc312 that can be aligned with at least one of the hands, where an outerdiameter of the secondary circular disc 312 forms the secondary circleand has a point aligned with the center of the hands. Thus, the maincomponents of the clock face 300 are generally similar to thosepresented in FIGS. 1 and 2, and similarly numbered elements can be thesame or similar to those described with respect to FIGS. 1 and 2.

The mechanical clock face 300 can generally have a motor 314 or othermechanism coupled to the hands 104, 106, 108 to create rotation aboutthe central axis 310 over time. The clock face 300 can comprise arelatively planar sheet located between the motor and the hands 104,106, 108, where the drive shafts 320 extend through the clock face 300and define the rotational axis of the hands 104, 106, 108. In thisconfiguration, the motor 314 would reside on the opposite side of theclock face 300 from the hands 104, 106, 108.

In this configuration, the clock face 300 can comprise axial markings inthe form of slits or openings 302, 304 in the clock face 300 that extendbetween the central axis 310 and the unit circle 102. In thisconfiguration, the clock face can support itself around the edge of theopenings 302, 304, for example, by being formed from a rigid materiallike metal or plastic. The clock drive shaft can be anchored by themotor and a motor plate which attaches to the clock face. In someembodiments, a portion of the clock face 300 may not be removed at ornear the central axis to provide stability to the clock face during use.The openings 302, 304 allow for the secondary circular disc 312 disposedbehind the clock face 300 to be viewed from the front of the clock face300.

In the example shown in FIGS. 3 and 4A, the secondary circular disc 312can be attached to the gearing for one of the hands such as the hourhand 108, where the secondary circular disc 312 has a diameter equal toa radius of the unit circle 102. While described herein as being usedwith the hour hand 108, the secondary circular disc 312 can be used withany of the hands 104, 106, 108, or any other suitable indicator. An edgeof the secondary circular disc 312 can be aligned with the axis ofrotation of the hour hand 108 such that rotation of the hour hand 108aligns an outer point of the secondary circular disc 312 substantiallywith the unit circle 102, though positioned on the opposite side of theclock face from the unit circle 102. The secondary circular disc 312 canbe formed of a disc of material (e.g., metal, plastic, glass, paper,etc.). While not shown, a background having a different color than thesecondary circular disc 312 can be present behind the secondary circulardisc 312 such that a contrasting color can allow the secondary circulardisc 312 to be easily viewed through the openings 302, 304. In thisarrangement, the secondary circular disc 312 can function as the sine orcosine indicator.

While shown as having the hands 104, 106, 108 on a side of the clockface 300, the clock face can, in some embodiments, be formed with atleast a portion having a transparent material. For example, the outerring past the unit circle 102 as shown in FIG. 4A can be transparent.One or more of the hands 104, 106, 108 can then be placed behind theclock face and show through the transparent portion. In theseembodiments, the unit circle 102 may be opaque, and only the portion ofthe hand 104, 106, 108 that extends beyond the unit circle 102 may bevisible. In some embodiments, a second backing may be present betweenthe clock face and the motor 314 so that only the hands between theclock face and the backing can be seen, which allows the second backingto conceal the motor and frame.

As shown in FIG. 4B, the secondary circular disc 312 can be disposed ona background disc 410 that can be symmetric about the central axis 310.In this embodiment, the secondary circular disc 312 can be attached toor disposed on the background disc 410. For example, the secondarycircular disc 312 can be printed on the background disc 410. In thisembodiment, the background disc 410 can have a radius that is equal toor greater than the radius of the openings 302, 304 such that thebackground disc 410 is visible through the openings. The background disc410 can then have the same color as the clock face so that the openingswould appear to only show the color of the secondary circular disc 312through the openings as the background disc 410 and the clock faceotherwise match. When the background disc 410 is symmetric about thecentral axis 310, the gearing of the clock may be less prone tounbalanced forces causing difficulties with the gearing and/orinaccuracies with keeping time. The background disc 410 can be coupledto and rotate with one of the hands 104, 106, 108. For example, thebackground disc 410 can be coupled to and rotate with the hour hand 108in some embodiments.

The clock display as shown in FIGS. 3 and 4A-4B can function similarlyto the clock display described with respect to FIG. 2. For example, thecosine can be displayed by providing an indication of the cosine valuecorresponding to a length of the secondary circular disc 312 visiblethrough the opening 304 for the x-axis, which corresponds to a length ofthe secondary circular disc 312 between the rotational axis and theintersection of the secondary circular disc 312 with the opening 304 forthe x-axis. The value of the cosine can range between −1 to 1, where azero length of the secondary circular disc 312 visible within theopening 304 corresponding to a value of zero for the cosine. A length ofthe secondary circular disc 312 visible within the opening 304 betweenthe center and the positive intersection of the opening 304 at the unitcircle 102 (e.g., the 3 O'clock position) can be defined to have a valueof 1, and a length of the secondary circular disc 312 visible within theopening 304 between the center and the negative intersection of theopening 304 at the unit circle 102 (e.g., the 9 O'clock position) can bedefined to have a value of −1. As shown in FIG. 3, the value of thecosine is illustrated by the relative amount of the secondary circulardisc 312 visible within the opening 304 between the rotational axistowards the 3 O'clock position.

Similarly, the sine can be displayed by providing an indication of thesine value corresponding to a length of the secondary circular disc 312visible through the opening 302 for the y-axis, which corresponds to alength of the secondary circular disc 312 visible between the rotationalaxis and the intersection of the secondary circular disc 312 with theopening 302 for the y-axis. The value of the sine can range between −1to 1, where a zero length of the secondary circular disc 312 visiblewithin the opening 302 corresponding to a value of zero for the sine. Alength of the secondary circular disc 312 visible within the opening 302between the center and the positive intersection of the opening 302 atthe unit circle 102 (e.g., the 12 O'clock position) can be defined tohave a value of 1, and a length of the secondary circular disc 312visible within the opening 302 between the center and the negativeintersection of the opening 302 at the unit circle 102 (e.g., the 6O'clock position) can be defined to have a value of −1. As shown in FIG.3, the value of the sine is illustrated by the relative amount of thesecondary circular disc 312 visible within the opening 302 between therotational axis towards the 6 O'clock position.

As shown in the embodiment of FIGS. 3 and 4A-4B, the display clock caninclude one or more hands to illustrate time, a unit circle encirclingthe hands, axial openings through a clock face aligned with the x-axisand the y-axis, and a secondary circular disc having a diameter equal toa radius of the unit circle. The secondary circular disc can have afirst point on a perimeter of the secondary circular disc aligned at thecenter point of the unit circle, and the secondary circular disc can bearranged to rotate about the first point. In this arrangement, an outerpoint on the secondary circular disc aligns with a point on the unitcircle. The sine and cosine functions can then be displayed based on anamount of the secondary circular disc visible through the axial openingsfor the x-axis and the y-axis. When used on a clock face, the secondarycircular disc can be coupled to one of the hands behind the clock faceto rotate about the center point during use of the clock, such that theclock can serve both to illustrate the time as well as the trigonometricfunctions of at least sine and cosine.

Other trigonometric functions can also be shown using the clocks asdescribed herein. Returning to FIG. 1, tangent and cotangent values canalso be displayed on any of the clocks described with respect to FIGS. 1to 4B. In this embodiment, tangent values appear as arrows on the leftand right sides of an enclosing square 150 and cotangent values asarrows on the top and bottom sides of the enclosing square 150. Thesquare can touch (e.g., by tangent to) the unit circle 102 at the 12O'clock, 3 O'clock, 6 O'clock, and 9 O'clock positions. Tangent valuesare ordinates (i.e. y-values) and cotangent values abscissae (i.e.x-values). Arrows start or end at zero. An arrow that points at zeroindicates its non-zero y or x value is to be negated. Thus tangents onthe left side of the square are the negative of the y coordinate valueand cotangents on the bottom of the square are the negative of the xcoordinate value. For example, if the hand angle is 30° or π/6≈0.52radians, then the tangent, t, is an arrow on the right side of thesquare from y=0 to y=1/(3)^(1/2)=(3)^(1/2)/3=1.732/3≈0.58; so t≈0.58. Ifthe angle is 150° or π−π/6=5π/6≈2.63 radians, then the arrow runs fromy=0.58 to y=0 on the left side of the square and it is the tangentt=−0.58. If the angle is 120° or 4π/6≈2.09 radians, then its cotangentis the arrow from x=0 to x=−1/(3)^(1/2)≈−0.58; so the cotangent ist⁻¹≈−0.58. If the angle is 240° or 8π/6≈4:19, then the cotangent is anarrow from x=−1/(3)^(1/2) to x=0 with value t⁻¹≈0.58. Tangent equalscotangent at the corners of the square such that the value is 1 if atthe upper right or lower left, and −1 if at the upper left or lowerright.

FIG. 5 illustrates still another embodiment of the clock display inwhich trigonometric summations can be visually displayed. The clock face500 is similar to that described with respect to FIGS. 1-4B except thatthe y-axis is labelled as imaginary so that the hands represent complexnumbers. In FIG. 5, similar numbers to those appearing in FIGS. 1-4B canrepresent the same or similar elements as described with respect toFIGS. 1-4B. In this arrangement, the minute hand 106 and the second hand104 can be multiplied as complex numbers to produce a third unit hand502, which manifests the trigonometric sum identities for cosine andsine. For example, the cosine and the sine are the arc sum of the minutehand's 106 arc 510 and the second hand's 104 arc 512. The three hands104, 106, 502 with their cosine and sine projections can produce ninehands, where a total of ten hands will appear when the hour hand 108 isincluded. In order to distinguish between the hands, the hands can havecontrasting colors, patterns, thicknesses, or the like on the display.

Multiplications of any two resulting hands produces an additional handwhose position is the sum of the arc lengths of the two arrows. Thisshows that the multiplication can be reduced to a summation. Moreover,this fact shows how to display DeMoivre's roots of unity on the unitcircle. DeMoivre's Theorem is equivalent to producing a unit hand or aunit arrow when multiplied by itself a number of times produces the unitarrow, namely the arrow pointing at (1,0) on the unit circle. As aresult, the unit arrow is a root of unity. The arrow opposite the unitarrow when multiplied by itself can produce the unity arrow. These twoarrows form a pair as the roots when there are two identical factors formultiplying. When there are three identical factors, the arrow pointingat 2π/3 will become the unit arrow since multiplying it once will put itat 2π/3+2π/3 and multiplying this arrow again will put it at2π/3+2π/3+2π/3=2π, which is the unit arrow. Likewise, the arrow pointingat 4π/3 when multiplied two more results in the unit arrow since4π/3+4π/3+4π/3=4π, the unit arrow. Thus for three identical factors, theroots are the arrows pointing at 2π/3, 4π/3 and 6π/3. This shows how tofind the root arrows, that they are equally spaced around the unitcircle, and that the number of distinct arrows equals the number ofidentical factors in the multiplication.

The clock can be used to perform a visual check of the trigonometric sumidentities of the terms of the component arrows. For example, as shownin FIG. 5, the clock face shows the time at 9 O'clock and 54 minutes.The minute hand arrow 106 is at 128 degrees with the arc 511 (e.g. thedark black arc in FIG. 5) having a length of 2.23. The second hand 104is at 36 seconds (e.g., 232 degrees) with an arc 512 having a length of4.05. The product of the minute and second hands 106, 104 is shown asthe arrow 502 is at 0 degrees (6.28 radians) with a total arc length asthe sum of the minute hand arc length of 2.23 and the second hand arclength of 4.05 for a total length of 6.28 (or 360 degrees). This arc isindicated by appending the minute hand black arc 511 to the second handarc 512 from 0.00 to 4.05 arriving at 6.28.

The sum of the arcs reflects the sum formulas of trigonometry, namely,the cosine of a sum of arcs is the difference of the hands cosines andsines (i.e., cos(α+β)=cos α cos β−sin α sin β) while the sine of a sumor arcs is the sum hands cosine and sine products (i.e., sin (α+β)=sin αcos β+cos α sin β). For the example as shown in FIG. 5, arrow502=1=arrow 504×arrow 508−arrow 506×arrow 514, and 0=arrow 506×arrow508+arrow 504×arrow 514, respectively. Thus, the clock provides a visualcheck of the trigonometric sum identities. If the second hand has an arcof 0 or 6.28, the product hand will overlay the minute hand. If thesecond hand has an arc of 3.14, the product hand will be opposite theminute hand. If the arc of the second hand is 1.57 or 4.71, the productwill be perpendicular to the minute hand. These allow for simplevisualization of the trigonometric sum formulation.

The sine and cosine clock, the sine and cosine clock with the tangentand cotangent functions, and/or the trigonometric summation clock can beimplemented on a computer. FIG. 6 illustrates a computer system 680suitable for implementing one or more embodiments of the display orclocks disclosed herein. The computer system 680 includes a processor682 (which may be referred to as a central processor unit or CPU) thatis in communication with memory devices including secondary storage 684,read only memory (ROM) 686, random access memory (RAM) 688, input/output(I/O) devices 690, and network connectivity devices 692. The processor682 may be implemented as one or more CPU chips.

It is understood that by programming and/or loading executableinstructions onto the computer system 680, at least one of the CPU 682,the RAM 688, and the ROM 686 are changed, transforming the computersystem 680 in part into a particular machine or apparatus having thenovel functionality taught by the present disclosure. It is fundamentalto the electrical engineering and software engineering arts thatfunctionality that can be implemented by loading executable softwareinto a computer can be converted to a hardware implementation bywell-known design rules. Decisions between implementing a concept insoftware versus hardware typically hinge on considerations of stabilityof the design and numbers of units to be produced rather than any issuesinvolved in translating from the software domain to the hardware domain.Generally, a design that is still subject to frequent change may bepreferred to be implemented in software, because re-spinning a hardwareimplementation is more expensive than re-spinning a software design.Generally, a design that is stable that will be produced in large volumemay be preferred to be implemented in hardware, for example in anapplication specific integrated circuit (ASIC), because for largeproduction runs the hardware implementation may be less expensive thanthe software implementation. Often a design may be developed and testedin a software form and later transformed, by well-known design rules, toan equivalent hardware implementation in an application specificintegrated circuit that hardwires the instructions of the software. Inthe same manner as a machine controlled by a new ASIC is a particularmachine or apparatus, likewise a computer that has been programmedand/or loaded with executable instructions may be viewed as a particularmachine or apparatus.

Additionally, after the system 680 is turned on or booted, the CPU 682may execute a computer program or application. For example, the CPU 682may execute software or firmware stored in the ROM 686 or stored in theRAM 688. In some cases, on boot and/or when the application isinitiated, the CPU 682 may copy the application or portions of theapplication from the secondary storage 684 to the RAM 688 or to memoryspace within the CPU 682 itself, and the CPU 682 may then executeinstructions that the application is comprised of In some cases, the CPU682 may copy the application or portions of the application from memoryaccessed via the network connectivity devices 692 or via the I/O devices690 to the RAM 688 or to memory space within the CPU 682, and the CPU682 may then execute instructions that the application is comprised of.During execution, an application may load instructions into the CPU 682,for example load some of the instructions of the application into acache of the CPU 682. In some contexts, an application that is executedmay be said to configure the CPU 682 to do something, e.g., to configurethe CPU 682 to perform the function or functions promoted by the subjectapplication. When the CPU 682 is configured in this way by theapplication, the CPU 682 becomes a specific purpose computer or aspecific purpose machine.

The secondary storage 684 is typically comprised of one or more diskdrives or tape drives and is used for non-volatile storage of data andas an over-flow data storage device if RAM 688 is not large enough tohold all working data. Secondary storage 684 may be used to storeprograms which are loaded into RAM 688 when such programs are selectedfor execution. The ROM 686 is used to store instructions and perhapsdata which are read during program execution. ROM 686 is a non-volatilememory device which typically has a small memory capacity relative tothe larger memory capacity of secondary storage 684. The RAM 688 is usedto store volatile data and perhaps to store instructions. Access to bothROM 686 and RAM 688 is typically faster than to secondary storage 684.The secondary storage 684, the RAM 688, and/or the ROM 686 may bereferred to in some contexts as computer readable storage media and/ornon-transitory computer readable media.

I/O devices 690 may include printers, video monitors, liquid crystaldisplays (LCDs), touch screen displays, keyboards, keypads, switches,dials, mice, track balls, voice recognizers, card readers, paper tapereaders, or other well-known input devices.

The network connectivity devices 692 may take the form of modems, modembanks, Ethernet cards, universal serial bus (USB) interface cards,serial interfaces, token ring cards, fiber distributed data interface(FDDI) cards, wireless local area network (WLAN) cards, radiotransceiver cards that promote radio communications using protocols suchas code division multiple access (CDMA), global system for mobilecommunications (GSM), long-term evolution (LTE), worldwideinteroperability for microwave access (WiMAX), near field communications(NFC), radio frequency identity (RFID), and/or other air interfaceprotocol radio transceiver cards, and other well-known network devices.These network connectivity devices 692 may enable the processor 682 tocommunicate with the Internet or one or more intranets. With such anetwork connection, it is contemplated that the processor 682 mightreceive information from the network, or might output information to thenetwork in the course of performing the above-described method steps.Such information, which is often represented as a sequence ofinstructions to be executed using processor 682, may be received fromand outputted to the network, for example, in the form of a computerdata signal embodied in a carrier wave.

Such information, which may include data or instructions to be executedusing processor 682 for example, may be received from and outputted tothe network, for example, in the form of a computer data baseband signalor signal embodied in a carrier wave. The baseband signal or signalembedded in the carrier wave, or other types of signals currently usedor hereafter developed, may be generated according to several methodswell-known to one skilled in the art. The baseband signal and/or signalembedded in the carrier wave may be referred to in some contexts as atransitory signal.

The processor 682 executes instructions, codes, computer programs,scripts which it accesses from hard disk, floppy disk, optical disk(these various disk based systems may all be considered secondarystorage 684), flash drive, ROM 686, RAM 688, or the network connectivitydevices 692. While only one processor 682 is shown, multiple processorsmay be present. Thus, while instructions may be discussed as executed bya processor, the instructions may be executed simultaneously, serially,or otherwise executed by one or multiple processors. Instructions,codes, computer programs, scripts, and/or data that may be accessed fromthe secondary storage 684, for example, hard drives, floppy disks,optical disks, and/or other device, the ROM 686, and/or the RAM 688 maybe referred to in some contexts as non-transitory instructions and/ornon-transitory information.

In some contexts, the secondary storage 684, the ROM 686, and the RAM688 may be referred to as a non-transitory computer readable medium or acomputer readable storage media. A dynamic RAM embodiment of the RAM688, likewise, may be referred to as a non-transitory computer readablemedium in that while the dynamic RAM receives electrical power and isoperated in accordance with its design, for example during a period oftime during which the computer system 680 is turned on and operational,the dynamic RAM stores information that is written to it. Similarly, theprocessor 682 may comprise an internal RAM, an internal ROM, a cachememory, and/or other internal non-transitory storage blocks, sections,or components that may be referred to in some contexts as non-transitorycomputer readable media or computer readable storage media.

Having described various devices and methods, specific embodiments caninclude, but are not limited to:

In a first embodiment, a trigonometric function display clock comprises:axial markings aligned with an x-axis and a y-axis of a clock face; anda secondary circle having a diameter equal to a radius of the axialmarkings between a center of each axial marking and an end of each axialmarking, wherein a first point on the secondary circle is aligned at thecenter of each axial marking, wherein the secondary circle is configuredto be rotatable about the first point, and wherein the secondary circleis configured to overlap and align with at least a portion of the axialmarkings as the secondary circle rotates about the first point.

A second embodiment can include the clock of the first embodiment,further comprising: a unit circle disposed on the clock face, wherein acenter of the unit circle is aligned with the center of each axialmarking, and wherein the center of the unit circle is aligned with acentral axis of the clock face.

A third embodiment can include the clock of the first or secondembodiment, wherein the secondary circle is disposed within the unitcircle, and wherein a second point on the secondary circle aligns withthe unit circle.

A fourth embodiment can include the clock of any one of the first tothird embodiments, further comprising: at least one hand configured torotate about the central axis.

A fifth embodiment can include the clock of the fourth embodiment,wherein the secondary circle is coupled to the at least one hand.

A sixth embodiment can include the clock of the fourth or fifthembodiment, wherein the at least one hand comprises an hour hand, aminute hand, and a second hand.

In a seventh embodiment, a trigonometric function display clockcomprises: axial openings in a clock face, wherein the axial openingsare aligned with an x-axis and a y-axis of the clock face; and asecondary circular disc having a diameter equal to a radius of eachaxial opening, wherein a first point on the secondary circular disc isaligned at a center of each axial marking, wherein the secondarycircular disc is configured to be rotatable about the first point,wherein the secondary circular disc is disposed on a first side of theclock face, and wherein the secondary circular disc is visible throughthe axial openings from a second side of the clock face.

An eighth embodiment can include the clock of the seventh embodiment,further comprising: a unit circle disposed on the clock face, wherein acenter of the unit circle is aligned with a central axis.

A ninth embodiment can include the clock of the seventh or eighthembodiment, further comprising: a background disc disposed on the secondside of the clock face, wherein the secondary circular disc is disposedon the background disc.

A tenth embodiment can include the clock of any one of the seventh toninth embodiments, further comprising: at least one hand configured torotate about the central axis, wherein the at least one hand is disposedon the second side of the clock face.

An eleventh embodiment can include the clock of the tenth embodiment,wherein the secondary circular disc is coupled to the at least one hand.

A twelfth embodiment can include the clock of the tenth or eleventhembodiment, wherein the at least one hand comprises an hour hand, aminute hand, or a second hand.

A thirteenth embodiment can include the clock of any one of the seventhto twelfth embodiments, wherein a second point on a perimeter of thesecondary circular disc aligns with the unit circle.

A fourteenth embodiment can include the clock of any one of the tenth tothirteenth embodiments, wherein the at least one hand is disposed on tehfirst side of the clock face.

In a fifteenth embodiment, a method of displaying one or moretrigonometric functions comprises: providing a display devicecomprising: a unit circle disposed on a clock face, wherein a center ofthe unit circle is aligned with a central axis; axial markings alignedwith an x-axis and a y-axis; and a secondary circle having a diameterequal to a radius of the unit circle, wherein a first point on thesecondary circle is aligned at the center of the unit circle, andwherein a second point on a perimeter of the secondary circle alignswith the unit circle; rotating the secondary circle about the firstpoint, wherein the secondary circle intersects with at least a portionof the axial markings as the secondary circle rotates about the firstpoint; and determining at least one of a value of a sine function or avalue of a cosine function at the second point based on the overlapbetween the secondary circle and the axial markings.

A sixteenth embodiment can include the method of the fifteenthembodiment, wherein determining the at least one of the value of thesine function or the cosine function at the second point comprises:determining a value of the cosine function at the second point, whereinthe value of the cosine function corresponds to a proportional lengthof: 1) a first length of the axial marking between the central axis andan intersection between the secondary circle and the axial marking onthe x-axis, and 2) a total length between the central axis and the unitcircle along the axial marking on the x-axis.

A seventeenth embodiment can include the method of the fifteenthembodiment, wherein determining the at least one of the value of thesine function or the cosine function at the second point comprises:determining a value of the sine function at the second point, whereinthe value of the sine function corresponds to a proportional lengthof: 1) a first length of the axial marking between the central axis andan intersection between the secondary circle and the axial marking onthe y-axis, and 2) a total length between the central axis and the unitcircle along the axial marking on the y-axis.

An eighteenth embodiment can include the method of the fifteenthembodiment, wherein the axial markings comprise openings in the clockface, wherein the secondary circle comprises a perimeter of a secondarycircular disc, and wherein the secondary circular disc, wherein thesecondary circular disc disposed on a first side of the clock face, andwherein the secondary circular disc is visible through the openings froma second side of the clock face.

A nineteenth embodiment can include the method of the eighteenthembodiment, wherein determining the at least one of the value of thesine function or the cosine function at the second point comprises:determining a value of the cosine function at the second point, whereinthe value of the cosine function corresponds to a proportional lengthof: 1) a first length of the secondary circular disc between the centralaxis and the unit circle visible through the opening on the x-axis, and2) a total length between the central axis and the unit circle along theopening on the x-axis.

A twentieth embodiment can include the method of the eighteenthembodiment, wherein determining the at least one of the value of thesine function or the cosine function at the second point comprises:determining a value of the sine function at the second point, whereinthe value of the sine function corresponds to a proportional lengthof: 1) a first length of the secondary circular disc between the centralaxis and the unit circle visible through the opening on the y-axis, and2) a total length between the central axis and the unit circle along theopening on the y-axis.

A twenty first embodiment can include the method of any one of thefifteenth to twentieth embodiments, wherein the display device furthercomprises: at least one hand configured to rotate about the centralaxis, wherein the at least one hand comprises an hour hand, a minutehand, and a second hand.

A twenty second embodiment can include the method of the twenty firstembodiment, wherein the secondary circle is coupled to the at least onehand.

A twenty third embodiment can include the method of the twenty first ortwenty second embodiment, further comprising: determining a time usingthe at least one hand on the display device.

A twenty fourth embodiment can include the method of any one of thefifteenth to twenty third embodiments, wherein the display devicefurther comprises an enclosing square displayed around the unit circle,wherein the method further comprises: determining at least one of atangent value or a cotangent value at the second point using theenclosing square.

In a twenty fifth embodiment, a trigonometric function display clockmethod comprises: executing a clock application on a processor, whereinthe clock application is stored in a memory, generating, by the clockapplication, a clock display having at least one hand indicator, whereinthe at least one hand indicator rotates about a central axis;generating, by the clock application, at least one of a cosine indicatoror a sine indicator on the clock display, wherein the at least one ofthe cosine indicator or the sine indicator is indicative of a value of acosine or sine, respectively, of a position of the at least one handindicator.

A twenty sixth embodiment can include the method of the twenty fifthembodiment, further comprising: generating and displaying a unit circleindicator on the clock display, wherein a center of the unit circleindicator is aligned with the central axis.

A twenty seventh embodiment can include the method of the twenty fifthor twenty sixth embodiment, further comprising: generating anddisplaying, by the clock application, axial markings aligned with anx-axis and a y-axis.

A twenty eighth embodiment can include the method of any one of thetwenty sixth to twenty seventy embodiments, further comprising:generating and displaying, by the clock application, a secondary circlehaving a diameter equal to a radius of the unit circle, wherein a firstpoint on the secondary circle is aligned at the center of the unitcircle, wherein the secondary circle is configured to be rotatable aboutthe first point, and wherein the secondary circle is configured tooverlap and align with at least a portion of the axial markings as thesecondary circle rotates about the first point.

A twenty ninth embodiment can include the method of any one of thetwenty sixth to twenty eighth embodiments, further comprising:generating and displaying, by the clock application, one or more arcindicators, wherein the one or more arc indicators are indicative of arotational distance of the at least one hand indicator from an originpoint.

A thirtieth embodiment can include the method of any one of the twentyfifth to twenty ninth embodiments, wherein the at least one handindicator comprises a plurality of hand indicators, and wherein at leasttwo hand indicators of the plurality of hand indicators are displayed indifferent colors.

While various embodiments in accordance with the principles disclosedherein have been shown and described above, modifications thereof may bemade by one skilled in the art without departing from the spirit and theteachings of the disclosure. The embodiments described herein arerepresentative only and are not intended to be limiting. Manyvariations, combinations, and modifications are possible and are withinthe scope of the disclosure. Alternative embodiments that result fromcombining, integrating, and/or omitting features of the embodiment(s)are also within the scope of the disclosure. Accordingly, the scope ofprotection is not limited by the description set out above, but isdefined by the claims which follow, that scope including all equivalentsof the subject matter of the claims. Each and every claim isincorporated as further disclosure into the specification and the claimsare embodiment(s) of the present invention(s). Furthermore, anyadvantages and features described above may relate to specificembodiments, but shall not limit the application of such issued claimsto processes and structures accomplishing any or all of the aboveadvantages or having any or all of the above features.

Additionally, the section headings used herein are provided forconsistency with the suggestions under 37 C.F.R. 1.77 or to otherwiseprovide organizational cues. These headings shall not limit orcharacterize the invention(s) set out in any claims that may issue fromthis disclosure. Specifically and by way of example, although theheadings might refer to a “Field,” the claims should not be limited bythe language chosen under this heading to describe the so-called field.Further, a description of a technology in the “Background” is not to beconstrued as an admission that certain technology is prior art to anyinvention(s) in this disclosure. Neither is the “Summary” to beconsidered as a limiting characterization of the invention(s) set forthin issued claims. Furthermore, any reference in this disclosure to“invention” in the singular should not be used to argue that there isonly a single point of novelty in this disclosure. Multiple inventionsmay be set forth according to the limitations of the multiple claimsissuing from this disclosure, and such claims accordingly define theinvention(s), and their equivalents, that are protected thereby. In allinstances, the scope of the claims shall be considered on their ownmerits in light of this disclosure, but should not be constrained by theheadings set forth herein.

Use of broader terms such as comprises, includes, and having should beunderstood to provide support for narrower terms such as consisting of,consisting essentially of, and comprised substantially of Use of theterm “optionally,” “may,” “might,” “possibly,” and the like with respectto any element of an embodiment means that the element is not required,or alternatively, the element is required, both alternatives beingwithin the scope of the embodiment(s). Also, references to examples aremerely provided for illustrative purposes, and are not intended to beexclusive.

While several embodiments have been provided in the present disclosure,it should be understood that the disclosed systems and methods may beembodied in many other specific forms without departing from the spiritor scope of the present disclosure. The present examples are to beconsidered as illustrative and not restrictive, and the intention is notto be limited to the details given herein. For example, the variouselements or components may be combined or integrated in another systemor certain features may be omitted or not implemented.

Also, techniques, systems, subsystems, and methods described andillustrated in the various embodiments as discrete or separate may becombined or integrated with other systems, modules, techniques, ormethods without departing from the scope of the present disclosure.Other items shown or discussed as directly coupled or communicating witheach other may be indirectly coupled or communicating through someinterface, device, or intermediate component, whether electrically,mechanically, or otherwise. Other examples of changes, substitutions,and alterations are ascertainable by one skilled in the art and could bemade without departing from the spirit and scope disclosed herein.

What is claimed is:
 1. A trigonometric function display clock, the clockcomprising: axial markings aligned with an x-axis and a y-axis of aclock face; and a secondary circle having a diameter equal to a radiusof the axial markings between a center of each axial marking and an endof each axial marking, wherein a first point on the secondary circle isaligned at the center of each axial marking, wherein the secondarycircle is configured to be rotatable about the first point, and whereinthe secondary circle is configured to overlap and align with at least aportion of the axial markings as the secondary circle rotates about thefirst point.
 2. The clock of claim 1, further comprising: a unit circledisposed on the clock face, wherein a center of the unit circle isaligned with the center of each axial marking, and wherein the center ofthe unit circle is aligned with a central axis of the clock face.
 3. Theclock of claim 1, wherein the secondary circle is disposed within theunit circle, and wherein a second point on the secondary circle alignswith the unit circle.
 4. The clock of claim 1, further comprising: atleast one hand configured to rotate about the central axis.
 5. The clockof claim 4, wherein the secondary circle is coupled to the at least onehand.
 6. The clock of claim 4, wherein the at least one hand comprisesan hour hand, a minute hand, and a second hand.
 7. A trigonometricfunction display clock, the clock comprising: axial openings in a clockface, wherein the axial openings are aligned with an x-axis and a y-axisof the clock face; and a secondary circular disc having a diameter equalto a radius of each axial opening, wherein a first point on thesecondary circular disc is aligned at a center of each axial marking,wherein the secondary circular disc is configured to be rotatable aboutthe first point, wherein the secondary circular disc is disposed on afirst side of the clock face, and wherein at least a portion of thesecondary circular disc is visible through the axial openings from asecond side of the clock face.
 8. The clock of claim 7, furthercomprising: a unit circle disposed on the clock face, wherein a centerof the unit circle is aligned with a central axis.
 9. The clock of claim7, further comprising: a background disc disposed on the second side ofthe clock face, wherein the secondary circular disc is disposed on thebackground disc.
 10. The clock of claim 7, further comprising: at leastone hand configured to rotate about the central axis, wherein the atleast one hand is disposed on the second side of the clock face.
 11. Theclock of claim 10, wherein the secondary circular disc is coupled to theat least one hand.
 12. The clock of claim 10, wherein the at least onehand comprises an hour hand, a minute hand, or a second hand.
 13. Theclock of claim 7, wherein a second point on a perimeter of the secondarycircular disc aligns with the unit circle.
 14. The clock of claim 10,wherein the at least one hand is disposed on the first side of the clockface.
 15. A method of displaying one or more trigonometric functions,the method comprising: providing a display device comprising: a unitcircle disposed on a clock face, wherein a center of the unit circle isaligned with a central axis; axial markings aligned with an x-axis and ay-axis; and a secondary circle having a diameter equal to a radius ofthe unit circle, wherein a first point on the secondary circle isaligned at the center of the unit circle, and wherein a second point ona perimeter of the secondary circle aligns with the unit circle;rotating the secondary circle about the first point, wherein thesecondary circle intersects with at least a portion of the axialmarkings as the secondary circle rotates about the first point; anddetermining at least one of a value of a sine function or a value of acosine function at the second point based on the overlap between thesecondary circle and the axial markings.
 16. The method of claim 15,wherein determining the at least one of the value of the sine functionor the cosine function at the second point comprises: determining avalue of the cosine function at the second point, wherein the value ofthe cosine function corresponds to a proportional length of: 1) a firstlength of the axial marking between the central axis and an intersectionbetween the secondary circle and the axial marking on the x-axis, and 2)a total length between the central axis and the unit circle along theaxial marking on the x-axis.
 17. The method of claim 15, whereindetermining the at least one of the value of the sine function or thecosine function at the second point comprises: determining a value ofthe sine function at the second point, wherein the value of the sinefunction corresponds to a proportional length of: 1) a first length ofthe axial marking between the central axis and an intersection betweenthe secondary circle and the axial marking on the y-axis, and 2) a totallength between the central axis and the unit circle along the axialmarking on the y-axis.
 18. The method of claim 15, wherein the axialmarkings comprise openings in the clock face, wherein the secondarycircle comprises a perimeter of a secondary circular disc, and whereinthe secondary circular disc, wherein the secondary circular discdisposed on a first side of the clock face, and wherein the secondarycircular disc is visible through the openings from a second side of theclock face.
 19. The method of claim 18, wherein determining the at leastone of the value of the sine function or the cosine function at thesecond point comprises: determining a value of the cosine function atthe second point, wherein the value of the cosine function correspondsto a proportional length of: 1) a first length of the secondary circulardisc between the central axis and the unit circle visible through theopening on the x-axis, and 2) a total length between the central axisand the unit circle along the opening on the x-axis.
 20. The method ofclaim 18, wherein determining the at least one of the value of the sinefunction or the cosine function at the second point comprises:determining a value of the sine function at the second point, whereinthe value of the sine function corresponds to a proportional lengthof: 1) a first length of the secondary circular disc between the centralaxis and the unit circle visible through the opening on the y-axis, and2) a total length between the central axis and the unit circle along theopening on the y-axis.
 21. The method of claim 15, wherein the displaydevice further comprises: at least one hand configured to rotate aboutthe central axis, wherein the at least one hand comprises an hour hand,a minute hand, and a second hand.
 22. The method of claim 21, whereinthe secondary circle is coupled to the at least one hand.
 23. The methodof claim 21, further comprising: determining a time using the at leastone hand on the display device.
 24. The method of claim 15, wherein thedisplay device further comprises an enclosing square displayed aroundthe unit circle, wherein the method further comprises: determining atleast one of a tangent value or a cotangent value at the second pointusing the enclosing square.
 25. The method of claim 15, wherein thedisplay device further comprises: at least one hand is configured torotate about about the central axis, wherein the at least one handcomprises a minute hand, wherein the secondary circle is coupled to theminute hand, wherein an hour hand is disposed on an opposite side of theclock face from the minute hand, and wherein the hour hand is visiblethrough at least a portion of the clock face.